0.7305 0 TD /F3 1 Tf 0 -2.363 TD 0.0001 Tc 20.6626 0 0 20.6626 413.829 701.0491 Tm /F2 1 Tf (+1)Tj /Length 6066 /F2 1 Tf 0 0 1 rg A set Cis midpoint convex if whenever two points a;bare in C, the average or midpoint (a+b)=2 is in C. Obviously a convex set is midpoint convex. 20.6626 0 0 20.6626 182.34 663.519 Tm >> 0 Tc (f)Tj /F7 1 Tf 20.6626 0 0 20.6626 255.204 541.272 Tm -21.4158 -1.2052 TD 20.6626 0 0 20.6626 501.255 518.709 Tm (H)Tj 0.2989 Tc [(\)L)300.5(e)250.3(t)]TJ /F1 4 0 R /F2 1 Tf [(eo)50.1(dory�s)-249.8(T)0.1(he)50.2(or)50.2(em,)-270.1(R)50.1(adon)100.1(�s)-249.8(The-)]TJ >> )]TJ /F2 1 Tf ()Tj (H)Tj /F2 1 Tf -21.7937 -1.2057 TD 0.0002 Tc /F4 1 Tf 0 Tc /ExtGState << 1.4827 0 TD /F4 1 Tf (b)Tj (I)Tj ()Tj 0 -1.2052 TD [(in)26(tersection)-332.9(o)-0.1(f)-333.2(a)-0.1(ll)-333.2(con)26(v)26.1(ex)-332.8(sets)-332.8(con)26(t)0(aining)]TJ 0.5798 0 TD /F1 1 Tf 4.0627 0 TD /F3 1 Tf 5.5102 0 TD /F2 1 Tf A set is convex i for any two points in the set their convex combination (line segment) is contained in the set. (=)Tj 20.6626 0 0 20.6626 453.762 626.313 Tm 414.25 625.823 l [(ane)-197.2(geometry:)]TJ /F4 7 0 R /F4 1 Tf 0.0001 Tc /F2 1 Tf 0.0001 Tc -19.3257 -1.2052 TD /F2 1 Tf 0.7836 0 TD S -6.969 -1.2052 TD /F5 1 Tf (,)Tj /F2 1 Tf [(the)-301.4(union)-301.9(of)-301.4(tetrahedra)-301.5(\(including)-301.9(in)26(terior)-301.4(p)-26.2(oin)26(ts\))-301.9(whose)]TJ /F4 1 Tf 11.3505 0 TD /F3 1 Tf 0.8947 0 TD 0 Tc 1.2087 0 TD ET 7.425 0 TD (C)Tj 0.7087 0 TD 1.1604 0 TD 0 Tc 387.355 636.114 l /F4 1 Tf (\))Tj /GS1 gs 1.782 0 TD /ExtGState << 3.3671 0 TD [(con)26.1(t)0.1(aining)]TJ 4.6 Convex Direction: Clearly every point in the convex set (shown in blue) can be the vertex for a ray with direction [1;0]T contained entirely in the convex set. 0 g /F4 1 Tf 0.0001 Tc 0.0041 Tc 0 Tc 0.3541 0 TD -14.6327 -1.2052 TD 4.8132 0 TD /F4 1 Tf /F4 1 Tf 0 -1.8712 TD (q)Tj [(,i)536.6(f)]TJ 14.3462 0 0 14.3462 153.135 638.9041 Tm 0.0001 Tc 1 i 0.8564 0 TD 20.6626 0 0 20.6626 241.857 267.4921 Tm 20.6626 0 0 20.6626 137.988 493.7971 Tm /F5 1 Tf -20.6884 -1.2052 TD 1 i [(only)-376.7(d)0(ep)-26.1(ends)-376.2(on)-376.8(the)]TJ (\))Tj stream )-813.2(In)-437.4(case)-437.3(1,)-471.2(assuming)]TJ /F3 1 Tf (:)Tj [(\))-342.3(s)0.1(uc)26.2(h)-343.2(t)0.1(hat)]TJ 0 Tc (|)Tj (cone\()Tj /Font << 0.7836 0 TD 226.093 685.464 200.694 710.863 169.4 710.863 c /F2 1 Tf (\))Tj 414.25 625.823 l /F3 1 Tf 14.3462 0 0 14.3462 295.29 674.175 Tm 42 0 obj /F4 1 Tf endobj This curve is not convex at all on the interval being graphed. (S)Tj 0 Tc /F4 1 Tf 2.3613 0 TD 414.25 625.823 m (I)Tj /F4 1 Tf 0.3337 0 TD /F4 1 Tf /F2 1 Tf /GS1 gs 20.6626 0 0 20.6626 365.445 493.7971 Tm /F2 1 Tf /F3 1 Tf 2.4898 0 TD 0.3337 0 TD /F2 1 Tf 14.3462 0 0 14.3462 196.461 375.2761 Tm 1.0763 0 TD . 0.2777 Tc 0.0001 Tc 4.1503 0 TD /F5 1 Tf 20.6626 0 0 20.6626 483.327 677.28 Tm -0.0001 Tc /F2 1 Tf (C)Tj stream 14.3462 0 0 14.3462 89.937 540.5161 Tm 0 g stream << 0.5101 0 TD The convex-hull operation is needed for the set of convex sets to form a lattice, in which the "join" operation is the convex hull of the union of two convex sets . 1.2113 0.95 TD /F5 1 Tf << (\()Tj /F2 1 Tf -21.7619 -1.2057 TD [(of)-359.4(dimen-)]TJ 0 Tc /F4 1 Tf -0.0002 Tc >> 0 Tc (i)Tj (\))Tj 0.4164 0 TD (. s 0.5893 0 TD (�s. (H)Tj ()Tj /F4 1 Tf [(,)-301.7(t)0.1(hat)-301.8(is,)]TJ (v)Tj (S)Tj /F5 1 Tf 387.657 636.416 l 0.8341 0 TD [(to)-452.2(the)-452.1(s)0.1(et)-452.1(of)-452.2(p)50(o)-0.1(sitive)-452.1(c)50.1(o)-0.1(mbinations)-451.6(o)-0.1(f)-451.8(families)-451.6(o)-0.1(f)]TJ /F1 1 Tf 0 Tc 14.3462 0 0 14.3462 431.712 526.593 Tm ()Tj -21.5619 -1.2052 TD (f)Tj 6.4502 0 TD 1.0903 0 TD 0.7836 0 TD 0.2775 Tc 0.3541 0 TD /F4 1 Tf 357.557 625.823 m 20.6626 0 0 20.6626 72 702.183 Tm /F3 1 Tf (�)Tj (S)Tj (i)Tj 27 0 obj /F4 1 Tf << [(,)-287.3(t)0.1(heorem)-283.7(3.2.2)-283.5(c)0.1(on�rms)-283.9(our)-283.5(in)26.1(tuition)-283.5(t)0.1(hat)]TJ 1.2 Convex set From the de nition of a ne sets, we can guess the de nition of convex sets. 0.5554 0 TD /F4 1 Tf [(,)-487.5(t)0.1(hen)-460.5(t)0.1(her)50.1(e)-460.8(exists)-460.3(a)-460.5(s)0.1(e)50.1(q)0(uenc)50.1(e)-460.8(o)-0.1(f)]TJ ()Tj 0 0 1 rg 14.3462 0 0 14.3462 374.274 404.769 Tm 14.3462 0 0 14.3462 353.682 587.3701 Tm [(c)50.1(onvex)-420.3(hul)-50.1(l)-420.4(of)]TJ 0.2777 Tc 20.6626 0 0 20.6626 221.58 541.272 Tm /Length 3049 0.8564 0 TD >> 0.446 Tc (. 0 0 1 rg ()Tj /F9 20 0 R /F4 1 Tf ()Tj 0 Tc The theory of convex functions is part of the general subject of convexity since a convex function is one whose epigraph is a convex set. /F4 1 Tf [(3.2. 20.6626 0 0 20.6626 417.555 258.078 Tm (m)Tj (> [(0)-917.3(f)0.1(or)-301.8(all)]TJ 0.0001 Tc /F4 1 Tf (\()Tj 0.3541 0 TD )-681.6(S)-0.1(ince)]TJ 4.7928 0 TD /F2 1 Tf /F3 1 Tf 8.4608 0 TD 0.0001 Tc 7.2429 0 TD [(con)26.1(v)-12.6(\()]TJ A set is called convex i any convex combination of a subset is also contained in the set itself. [(\),)-423.8(but)-399(if)]TJ -9.6165 -2.3625 TD [(,o)273(r)]TJ /F5 1 Tf 0.0002 Tc 2.0324 0 TD 1.143 0 TD 20.6626 0 0 20.6626 232.173 292.4041 Tm (S)Tj [(CHAPTER)-327.3(3. 21.1364 0 TD /F2 1 Tf 0 Tc 442.597 654.17 m Convex hull: set of all convex combination of points in S. Denotes as Conv(S). [(denoted)-446.1(b)26(y)]TJ /F2 1 Tf ()Tj Formally, if X:= fx i 2Rn j1 i mgis an arbitrary set of points, then its convex hull is the set obtained by taking all possible convex combinations of the points in X. (\(with)Tj 0.5893 0 TD /F2 1 Tf )]TJ )-435.6(F)74.9(or)-306.5(any)-306.8(p)50.1(oint,)]TJ 0 Tc [(The)-247.9(e)0(mpt)26.1(y)-247.9(set)-248.3(is)-248.3(trivially)-248.3(con)26(v)26.1(ex,)-258.7(e)0(v)26.1(ery)-247.9(one-p)-26.2(oin)26(t)-247.8(set)]TJ /F4 1 Tf /F1 1 Tf [(con)26.1(v)-13(\()]TJ /F2 1 Tf /F5 1 Tf (S)Tj (of)Tj So, any shape which is concave, or has a hollow, cannot be a convex set. /F5 1 Tf /F2 5 0 R /F4 1 Tf (\))Tj 357.557 597.477 m (. BT [(spanned)-266.1(b)26.1(y)]TJ )-590.1(Giv)26.1(e)0(n)-363.4(a)-0.1(n)26(y)-362.9(set)-363.3(of)-362.8(v)26.1(ectors,)]TJ 442.597 597.477 m 0 Tc [(\))-350(i)0(s)-350(t)0.2(he)-349.6(c)50.2(onvex)-350.1(hul)-50(l)-350.1(of)]TJ [(of)-301.8(strictly)-301.9(less)-301.9(than)]TJ [(\(2\))-301.4(I)0(s)-301.4(i)0(t)-301(n)-0.1(ecessary)-301.5(to)-301(consider)-301.4(con)26(v)26.1(ex)-301.1(com)25.9(b)-0.1(inations)-301(of)-301.4(all)]TJ 1.0437 0 TD /F4 1 Tf -0.0001 Tc 0 Tc 0 g /F5 1 Tf 11.7569 0 TD T* (I)Tj /F2 1 Tf 7.6254 0 TD 0.3938 Tc 20.3985 0 TD ()Tj /F4 1 Tf 1.0528 0 TD 0.0001 Tc (of)Tj )Tj 4.4007 0 TD 1.4958 0 TD T* BT (i)Tj 11.9551 0 0 11.9551 72 736.329 Tm 13.4618 0 TD 0.0001 Tc 0 Tc [(and)-420.4(th)26(us,)-449.9(a)-420.8(set)]TJ /F5 1 Tf )Tj 0.3499 Tc 0.0001 Tc Preface The theory of convex sets is a vibrant and classical field of modern mathe-matics with rich applications in economics and optimization. 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